On Deduction (1950-1974)

"Insight is not the same as scientific deduction, but even at that it may be more reliable than statistics." (Anthony Standen, "Science Is a Sacred Cow", 1950)

"[…] the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather from them, than for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." (Alfred N Whitehead, "Principia Mathematica", 1950)

"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors." (Richard von Mises, "Positivism: A Study in Human Understanding", 1951)

"We are driven to conclude that science, like mathematics, is a system of axioms, assumptions, and deductions; it may start from being, but later leaves it to itself, and ends in the formation of a hypothetical reality that has nothing to do with existence; or it is the discovery of an ideal being which is, of course, present in what we call actuality, and renders it an existence for us only by being present in it." (Poolla T Raju, "Idealistic Thought of India", 1953)

"[…] the grand aim of all science […] is to cover the greatest possible number of empirical facts by logical deductions from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"This method of deduction [...] is often called 'combinatory'. Its usefulness is not exhausted at this stage, but it does even at the outset lead to some valuable conclusions [...]" (John Chadwick, "The Decipherment of Linear B", 1958)

"The functional validity of a working hypothesis is not a priori certain, because often it is initially based on intuition. However, logical deductions from such a hypothesis provide expectations (so called prognoses) as to the circumstances under which certain phenomena will appear in nature. Such a postulate or working hypothesis can then be substantiated by additional observations or by experiments especially arranged to test details. The value of the hypothesis is strengthened if the observed facts fit the expectation within the limits of permissible error." (R Willem van Bemmelen, "The Scientific Character of Geology", The Journal of Geology Vol 69 (4), 1961)

"Intellect begins with the observation of nature, proceeds to memorize and classify the facts thus observed, and by logical deduction builds up that edifice of knowledge properly called science. But admittedly we also know by feeling, and we can combine the two faculties, and present knowledge in the guise of art." (Herbert Read, "Selected Writings: Poetry and Criticism", 1963)

"Perfect logic and faultless deduction make a pleasant theoretical structure, but it may be right or wrong; the experimenter is the only one to decide, and he is always right." (Léon Brillouin, "Scientific Uncertainty and Information", 1964)

"The interplay between generality and individuality, deduction and construction, logic and imagination - this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be found at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the 'concrete', then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy: after this flight comes the crucial test for learning and reaching specific goals in the newly surveyed low plains of individual 'reality'. In brief, the flight into abstract generality must start from and return again to the concrete and specific." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964) 

"So in order to understand the physics one must always have a neat balance and contain in his head all of the various propositions and their interrelationships because the laws often extend beyond the range of their deductions. This will only have no importance when all the laws are known." (Richard Feynman, "The Character of Physical Law", 1965)

"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)

"Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties." (Herbert Dingle, "Science at the Crossroads", 1972)

"To give a causal explanation of an event means to deduce a statement which describes it, using as premises of the deduction one or more universal laws, together with certain singular statements, the initial conditions. [...] We have thus two different kinds of statement, both of which are necessary ingredients of a complete causal explanation." (Karl Popper, "The Philosophy of Karl Popper", 1974)